Don't individual signal sources affect each other when using a summing amplifier?

I cannot intuitively see why this circuit works.

Consider a somewhat ideal op-amp with an open loop gain of (say) 100,000. Consider next that the output voltage at any point in time is not saturated against the power rails i.e. the circuit is behaving like a linear amplifier. Then, imagine the output voltage, at any particular point in time was (say) 10 volts. This MUST mean that the voltage difference between inverting and non-inverting inputs is 10/100,000 = 0.1 mV.

And, if the non-inverting input is tied to 0 volts, then the inverting input is at about 0.1 mV. If the output signal was a peak-to-peak signal between -10 volts and +10 volts then the non-inverting input will change +/-0.1 mV accordingly.

This is why we call it a virtual earth; summing amplifiers make use of this to add the currents from each input source because adding voltages directly is problematic.

If you factored in the +/- 0.1 mV change theoretically there is a slight influence from one input voltage of the mixer on another input but, it is negligible.

Smallprint: It's only negligible if the real op-amp used has decent open-loop gain throughout the bandwidth of the signal. So, for instance, if the op-amp chosen has an open-loop DC gain of 1,000,000 it might only have an open-loop gain of 100,000 at 10 Hz. Taking this further, the open-loop gain might be only 100 at 10 kHz.

So, to produce a +/- 10 volt sinewave at 10 kHz, the difference signal on the inverting input is +/-0.1 volts and not the piddling amount at DC.

This is why real, quality, op-amp mixers use op-amps that are far superior to what might be initially felt to be needed.

The answer is in the schematic you provided. The inverting input is a "virtual earth".

The amplifier is configured so that the output adjusts until the negative feedback causes the inverting input to match the non-inverting. Since the non-inverting input is tied to GND the inverting input is driven close enough to 0 V that we call it "virtual ground".

^{simulate this circuit – Schematic created using CircuitLab}

Figure 1. The equivalent circuit as seen by the input signals.

As shown, there is no cause for the signals to interfere with each other.